An inverse problem for a quasilinear convection-diffusion equation (Q2145622)

Let \(T > 0\) and \(\Omega \subset \mathbb{R}^n\), \(n\geq 2\) a bounded domain with a smooth boundary. The authors consider the problem of determining the nonlinear diffusion coefficient \(a(t,\xi)\) and the nonlinear convection term \(B\) in the initial boundary problem \[ \left\{ \begin{array}{ll} \partial_t u - \operatorname{div}(a(t,u)\nabla u) - B(t,x,u,\nabla u)\cdot \nabla u = 0 \text{ in } (0,T) \times \Omega,\\ u = \lambda + f \text{ on } (0,T)\times\partial\Omega,\\ u(0,x) = \lambda \text{ in } \Omega \end{array} \right. \] from the knowledge of the parabolic Dirichlet-to-Neumann map for all \(\lambda \in \mathbb{R}\). The main result states that, under suitable assumptions, it is possible to fully recover \(a\) and \(B\).